Kurtosis

The degree of peakedness of a distribution, also called the Excess or Excess Coefficient. Kurtosis is denoted $\gamma_2$ (or $b_2$) or $\beta_2$ and computed by taking the fourth Moment of a distribution. A distribution with a high peak $(\gamma_2 > 0)$ is called Leptokurtic, a flat-topped curve $(\gamma_2 < 0)$ is called Platykurtic, and the normal distribution $(\gamma_2 = 0)$ is called Mesokurtic. Let $\mu_i$ denote the $i$th Moment $\left\langle{x^i}\right\rangle{}$. The Fisher Kurtosis is defined by

$$\gamma_2 \equiv b_2 \equiv {\mu_4\over {\mu_2}^2} - 3 = {\mu_4\over \sigma^4} - 3,$$ (1)

and the Pearson Kurtosis is defined by

$$\beta_2 \equiv \alpha_4 \equiv {\mu_4\over \sigma^4}.$$ (2)

An Estimator for the $\gamma_2$ Fisher Kurtosis is given by

$$g_2 = {k_4\over {k_2}^2},$$ (3)

where the $k$s are k-Statistic. The Standard Deviation of the estimator is

$${\sigma_{g_2}}^2 \approx {24\over N}.$$ (4)

See also Fisher Kurtosis, Mean, Pearson Kurtosis, Skewness, Standard Deviation

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 928, 1972.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Moments of a Distribution: Mean, Variance, Skewness, and So Forth.'' §14.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 604-609, 1992.